Example
In driving in a straight line from New York to Boston,
your position function given in miles from New York is described by the
function:

where t is the number of hours
since the trip began. It takes you 8 hours to get to Boston.

a) Find your
velocity at time t= 1/2 hrs.

b) Do you ever backtrack during the trip?

Solution to a): First, we find your velocity function at any time by
differentiating your position
function.

v(t) = f'(t) = 5 t2 - 50 t + 120.

Then, to find your velocity after half an hour, we just plug 1/2 into the
velocity function.

Hey, it sounds high but a lot of states have upped their speed limits.

Solution to b) (Do you backtrack?): Yes you do. It seems you left your favorite Mets baseball cap at
the rest stop when you
took it off to check your thinning hair in the mirror. Passing through
Providence, you noticed the shiny spot on your head when you looked in the
rear view
mirror and realized you had forgotten it. Unfortunately, when you got back
to the rest stop, you found it in the toilet, where it had been left by a
Phillies fan.

How do we see that you backtracked? Well, if the velocity function is ever
negative during the trip (that is, when
), then you must have
been backtracking at that point. But notice that

v(t)= 5 (t2 - 10 t + 24) = 5(t-6) (t-4).

In particular, v(t) = 0 when t= 4 hrs and t=6 hrs. That would be
when you realized you lost your hat and turned around, making your velocity
go from positive to negative, and also when you turned around at the rest
stop after you realized that you could flush the hat goodbye. So we would
expect that in between those times, you were backtracking and your velocity
was negative. Just to be sure, let's check what the velocity was at time
t = 5.
Then,

Yup, it's negative. You definitely backtracked. You really wanted
that hat. But you probably could have made better time if you weren't
backing down the shoulder of the northbound lanes. Oh, well....